- #1
Michael_0039
- 59
- 7
- Homework Statement
- I used D'alembert criterion to determine if the mathematical sequence converges or diverges
- Relevant Equations
- Σ(1/(n+1)^2
What is the solution to this mathematical series of real numbers?
- Thread starter Michael_0039
- Start date
Click For Summary
Homework Help Overview
The discussion revolves around a mathematical series of real numbers, specifically focusing on its convergence properties and limit evaluation. Participants are exploring various approaches to analyze the series.
Discussion Character
- Exploratory, Mathematical reasoning, Assumption checking
Approaches and Questions Raised
- Some participants discuss the limit of a specific expression and its implications for the series' convergence. Others suggest comparing the series to known convergent series and mention various convergence tests, including the Cauchy condensation criterion and the integral test. The original poster expresses interest in using D'alembert's criterion.
Discussion Status
The discussion includes multiple approaches to understanding the series' convergence, with participants offering different methods and criteria for analysis. There is no explicit consensus on a single method, but several productive lines of reasoning are being explored.
Contextual Notes
Participants are working within the constraints of mathematical series convergence and are referencing established results and criteria without providing complete solutions. The original poster's intent to use D'alembert's criterion indicates a focus on exploring various methods of analysis.
- #2
Delta2
Homework Helper
- 6,002
- 2,628
You have a mistake in the last line, it is easy to prove that ##\lim_{n \to +\infty}(1-\frac{1}{n+2})^{n+2}=e^{-1}=\lim_{n \to +\infty}(1-\frac{1}{n+2})^{n}## so the final limit is 1.
Instead upper bound the series by the hyperharmonic series (p-series for p=2 )which is a well know result that it converges, hence by the comparison criterion this series converges too.
Instead upper bound the series by the hyperharmonic series (p-series for p=2 )which is a well know result that it converges, hence by the comparison criterion this series converges too.
- #3
member 587159
Comparing with the convergent series
$$\sum_{n=1}^\infty \frac{1}{n^2} =\frac{\pi^2}{6}$$ yields the result.
Alternatively, use the Cauchy condensation criterium or the integral test.
$$\sum_{n=1}^\infty \frac{1}{n^2} =\frac{\pi^2}{6}$$ yields the result.
Alternatively, use the Cauchy condensation criterium or the integral test.
- #4
Michael_0039
- 59
- 7
Thanks for your answer. I wanted to use D'alembert criterion to see if a solution can be found.
Similar threads
- · Replies 1 ·
- · Replies 3 ·
- · Replies 8 ·
- · Replies 7 ·
- · Replies 6 ·